Hasan Rowhani's stunning presidential election victory in Iran has opened the door for improved relations with the West, but the U.S. and Israel remain cautious about making progress on their key demand — dismantling Iran's nuclear program.

Israeli Prime Minister Benjamin Netanyahu on Sunday said the international community "must not give in to wishful thinking or temptation and loosen the pressure on Iran for it to stop its nuclear program."

And the White House, in a statement released shortly after Rowhani's victory was announced Saturday, offered only cautious support, saying the United States "remains ready to engage the Iranian government directly in order to reach a diplomatic solution that will fully address the international community's concerns about Iran's nuclear program."

For his part, Rowhani, who trounced a field of otherwise hard-line presidential candidates, has said his election is "an opportunity in the international scene for those who support democracy to speak to this great nation through respectful and fair language while admitting its rights."

And, in any case, Rowhani is likely to be friendlier to the West than his combative predecessor, Mahmoud Ahmadinejad.

His options are limited, however.

Rowhani, who doesn't take office until August, has little control over the country's military and nuclear programs, which fall under the purview of the country's ruling Islamic clerics. But the president-elect will be in charge of key aspects of the economy, which has been crippled by Western sanctions aimed at preventing Tehran from acquiring nuclear weapons.

On Sunday, Rowhani warned that Iran's 30 percent inflation and 14 percent unemployment rates could not be reversed "overnight."

Meanwhile, ordinary Iranians, who turned out in large numbers to push Rowhani to victory, celebrated into Sunday, reveling in their repudiation of the country's conservative hard-liners.

"Riot police, who were frequently deployed on Tehran streets in the run-up to Friday's vote, were conspicuous in their absence. State TV showed footage of the celebrations and rebroadcast a speech he made after his victory was announced Saturday, asserting Iran's readiness to improve its ties with the world.

"So far, Iran's establishment seems also to have accepted the Rowhani victory. The country's powerful Revolutionary Guard announced on its webpage its "comprehensive readiness for interaction and cooperation with the next administration in the framework of legal duties and assignments."